Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

• Published in: Numerische Mathematik Vol. 131; no. 3; pp. 567 - 614
• Main Authors: Gander, M. J., Graham, I. G., Spence, E. A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2015
• Subjects: Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Simulation; Theoretical; 78A45; 35J05; 65F10; 65F08; 65N30

There has been much recent research on preconditioning discretisations of the Helmholtz operator Δ + k 2 (subject to suitable boundary conditions) using a discrete version of the so-called “shifted Laplacian” Δ + ( k 2 + i ε ) for some ε > 0 . This is motivated by the fact that, as ε increases, t...